[lupire]

A better example is the "real" numbers, which do not physically exist but are a constant source of theorems that don't apply to physics, and this is taken as proof that real numbers are a fascinating complex object, instead of a poor foundation for analysis. Standard Mathematics does not use a good model for infinitesimals, which should not be allowed to be densely non-differentiable.

[raverbashing]

Yep, this is a good example. And maybe it helps illustrating my point

On N and Q we can come up with any number we want. Even though there are infinite numbers, we can come up with any number

On R we can do that as well. But at the same time we can come with "vapid" statements like "for each x in R exists x' < x" which are completely correct but don't really mean anything.

True, the R's are a good foundation, and in a way they do exist physically (after all, Pi exists in nature).